LINEAR THEORY OF FLUCTUATIONS 121 



where R and c are the average^^ values of R{l) and c{r, t) respectively. 

 Evidently, Ri{t) = 0, and Ci(r, /) = 0. Introducing the expressions (2.2) 

 and (2.3) into Eq. (2.1), expanding in terms of Ci(r, /), and neglecting terms 

 of the order of Ci, we get 



Ri{l) = f m cr(r, I) dr, (2.4) 



where 



The function /(r) defines the linear coupling between Ri and Ci and depends 

 upon the specific physical model used. The non-linear terms neglected in 

 Eq. (2.4) may be of importance under some conditions; however, we will 

 not consider them here. Nevertheless, non-linear effects in the behavior of 

 Ci itself are possibly important in determining the form of the power spec- 

 trum of Ri{l) in the neighborhood of zero frequency. 



3. The Fourier Series Method of Solution 



In this section we consider the state of the diffusing system to be defined 

 by the Fourier space-amplitudes Ck(i) of Ci(r, /)• The time behavior of Ck{t) 

 will be described by an infinite set of ordinary differential equations con- 

 taining random exciting forces according to the conventional theory of 

 Brownian motion.^^ This method yields the spectral density of Ri(l) directly. 



Now the diffusion process is assumed to occur in a rectangular area A2 = 

 Li X L2 or in a rectangular parallelopiped of volume ^3 = Zi X ^-2 X Z-s • 

 In regions of the above types, if we apply periodic boundary conditions , 

 ci(ro/) may be expanded in Fourier space-series as follows: 



c,{r, t) = E' CkiDe'"'-' (3.1) 



k 



where the components of k take the values 



ki = linti/Li , i = \, • • ■ ,v, (3.2) 



in which rii are integers and v is the number of dimensions. The prime on 

 the summation indicates that the term for fe = is to b3 omitted. This is 

 required by the equivalence of the time and space averages of Ci (true for 

 A, sufficiently large) and by the vanishing of the time average of Ci (by 

 definition). 



'^ The average values here may be considered as either time or ensemble averages but 

 not space averages. 



" See Wang and Uhlenbeck, Rev. Mod. Pliys., 17, 323-342 (1945). 



" If the final results are given by integrals over k- space they will be insensitive to the 

 boundary conditions. 



